Covariance structure analysis: statistical practice, theory, and directions.

Although covariance structure analysis is used increasingly to analyze nonexperimental data, important statistical requirements for its proper use are frequently ignored. Valid conclusions about the adequacy of a model as an acceptable representation of data, which are based on goodness-of-fit test statistics and standard errors of parameter estimates, rely on the model estimation procedure being appropriate for the data. Using analogies to linear regression and anova, this review examines conditions under which conclusions drawn from various estimation methods will be correct and the consequences of ignoring these conditions. A distinction is made between estimation methods that are either correctly or incorrectly specified for the distribution of data being analyzed, and it is shown that valid conclusions are possible even under misspecification. A brief example illustrates the ideas. Internet access is given to a computer code for several methods that are not available in programs such as EQS or LISREL.

[1]  R S Lockhart Asymptotic sampling variances for factor analytic models identified by specified zero parameters , 1967, Psychometrika.

[2]  K. Jöreskog A general approach to confirmatory maximum likelihood factor analysis , 1969 .

[3]  M. Browne Generalized Least Squares Estimators in the Analysis of Covariance Structures. , 1973 .

[4]  P. Krishnaiah,et al.  On Covariance Structures. , 1975 .

[5]  William R. Krane,et al.  Scale invariance and the factor analysis of correlation matrices , 1978 .

[6]  P. Bentler MULTIVARIATE ANALYSIS WITH LATENT VARIABLES: CAUSAL MODELING , 1980 .

[7]  D. G. Weeks,et al.  Linear structural equations with latent variables , 1980 .

[8]  Bentler,et al.  Some asymptotic properties of constrained generalized least squares estimation in coy ariance structure models , 1980 .

[9]  Taeke Klaas Dijkstra Latent variables in linear stochastic models , 1981 .

[10]  G. Chamberlain Multivariate regression models for panel data , 1982 .

[11]  Michael W. Browne,et al.  Topics in Applied Multivariate Analysis: COVARIANCE STRUCTURES , 1982 .

[12]  Alexander Shapiro,et al.  Asymptotic distribution theory in the analysis of covariance structures , 1983 .

[13]  P. Bentler Some contributions to efficient statistics in structural models: Specification and estimation of moment structures , 1983 .

[14]  P. Bentler,et al.  Simultaneous equation systems as moment structure models : With an introduction to latent variable models , 1983 .

[15]  A. Boomsma On the robustness of LISREL (maximum likelihood estimation) against small sample size and non-normality. , 1984 .

[16]  D. A. Kenny,et al.  Estimating the nonlinear and interactive effects of latent variables. , 1984 .

[17]  C. Raghavendra Rao,et al.  Multivariate Analysis and Its Applications , 1984 .

[18]  B. Muthén A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators , 1984 .

[19]  R. P. McDonald,et al.  Some algebraic properties of the Reticular Action Model for moment structures. , 1984, The British journal of mathematical and statistical psychology.

[20]  M. Browne Asymptotically distribution-free methods for the analysis of covariance structures. , 1984, The British journal of mathematical and statistical psychology.

[21]  Ab Mooijaart,et al.  Factor analysis for non-normal variables , 1985 .

[22]  Sik-Yum Lee,et al.  On testing functional constraints in structural equation models , 1985 .

[23]  W. Newey,et al.  Large sample estimation and hypothesis testing , 1986 .

[24]  C Loehlin John,et al.  Latent variable models: an introduction to factor, path, and structural analysis , 1986 .

[25]  J. S. Tanaka "How Big Is Big Enough?": Sample Size and Goodness of Fit in Structural Equation Models with Latent Variables. , 1987 .

[26]  A. Shapiro,et al.  Analysis of Covariance Structures under Elliptical Distributions , 1987 .

[27]  P. Bentler,et al.  Multiple population covariance structure analysis under arbitrary distribution theory , 1987 .

[28]  M. Browne Robustness of statistical inference in factor analysis and related models , 1987 .

[29]  T. W. Anderson,et al.  The asymptotic normal distribution of estimators in factor analysis under general conditions , 1988 .

[30]  Jan de Leeuw,et al.  Multivariate analysis with linearizable regressions , 1988 .

[31]  A. Shapiro,et al.  Robustness of normal theory methods in the analysis of linear latent variate models. , 1988 .

[32]  Kenneth A. Bollen,et al.  Structural Equations with Latent Variables , 1989 .

[33]  T. Micceri The unicorn, the normal curve, and other improbable creatures. , 1989 .

[34]  Karl G. Jöreskog,et al.  Lisrel 8: User's Reference Guide , 1997 .

[35]  R. P. McDonald,et al.  An index of goodness-of-fit based on noncentrality , 1989 .

[36]  A. Satorra Alternative test criteria in covariance structure analysis: A unified approach , 1989 .

[37]  Bengt Muthén,et al.  MULTILEVEL ASPECTS OF VARYING PARAMETERS IN STRUCTURAL MODELS , 1989 .

[38]  Ronald Schoenberg,et al.  Pseudo maximum likelihood estimation and a test for misspecification in mean and covariance structure models , 1989 .

[39]  T. W. Anderson,et al.  Asymptotic Chi-Square Tests for a Large Class of Factor Analysis Models , 1990 .

[40]  S. Breckler Applications of covariance structure modeling in psychology: cause for concern? , 1990, Psychological bulletin.

[41]  S. Breckler,et al.  Applications of covariance structure modeling in psychology: cause for concern? , 1990, Psychological bulletin.

[42]  Albert Satorra,et al.  Model Conditions for Asymptotic Robustness in the Analysis of Linear Relations , 1990 .

[43]  Sik-Yum Lee,et al.  Multilevel analysis of structural equation models , 1990 .

[44]  S. Hershberger,et al.  A Simple Rule for Generating Equivalent Models in Covariance Structure Modeling. , 1990, Multivariate behavioral research.

[45]  Peter M. Bentler,et al.  Covariance structure analysis with heterogeneous kurtosis parameters , 1990 .

[46]  A. Satorra,et al.  Scaled test statistics and robust standard errors for non-normal data in covariance structure analysis: a Monte Carlo study. , 1991, The British journal of mathematical and statistical psychology.

[47]  P. Bentler,et al.  Robustness of normal theory statistics in structural equation models , 1991 .

[48]  R. Cudeck,et al.  Model selection in covariance structures analysis and the "problem" of sample size: a clarification. , 1991 .

[49]  P. Bentler,et al.  Covariance Structure Analysis Under a Simple Kurtosis Model , 1992 .

[50]  Yutaka Kano,et al.  Robust statistics for test-of-independence and related structural models , 1992 .

[51]  R. MacCallum,et al.  Model modifications in covariance structure analysis: the problem of capitalization on chance. , 1992, Psychological bulletin.

[52]  Y Kano,et al.  Can test statistics in covariance structure analysis be trusted? , 1992, Psychological bulletin.

[53]  Albert Satorra,et al.  Asymptotic robust inferences in the analysis of mean and covariance structures , 1992 .

[54]  B. Muthén,et al.  A comparison of some methodologies for the factor analysis of non‐normal Likert variables: A note on the size of the model , 1992 .

[55]  Peter M. Bentler,et al.  Structural equation models with continuous and polytomous variables , 1992 .

[56]  Duane T. Wegener,et al.  The problem of equivalent models in applications of covariance structure analysis. , 1993, Psychological bulletin.

[57]  B. Everitt,et al.  Modelling Covariances and Latent Variables Using EQS , 1993 .

[58]  J. S. Long,et al.  Testing Structural Equation Models , 1993 .

[59]  Robert Cudeck,et al.  A simple Gauss-Newton procedure for covariance structure analysis with high-level computer languages , 1993 .

[60]  D. Wegener,et al.  The problem of equivalent models in applications of covariance structure analysis. , 1993, Psychological bulletin.

[61]  L. Harlow,et al.  Causation issues in structural equation modeling research , 1994 .

[62]  P. Dudgeon A reparameterization of the restricted factor analysis model for multitrait‐multimethod matrices , 1994 .

[63]  Tom Wansbeek,et al.  Identification, Equivalent Models, and Computer Algebra , 1994 .

[64]  John B. Willett,et al.  Using covariance structure analysis to detect correlates and predictors of individual change over time , 1994 .

[65]  A. Satorra,et al.  Corrections to test statistics and standard errors in covariance structure analysis. , 1994 .

[66]  P M Bentler,et al.  Bootstrap-corrected ADF test statistics in covariance structure analysis. , 1994, The British journal of mathematical and statistical psychology.

[67]  Karl G. Jöreskog,et al.  Structural equation modeling with ordinal variables , 1994 .

[68]  Ingram Olkin,et al.  MULTIVARIATE NON-NORMAL DISTRIBUTIONS AND MODELS OF DEPENDENCY , 1994 .

[69]  P. Bentler,et al.  A Note on Using and Unbiased Weight Matrix in the ADF Test Statistic. , 1995, Multivariate behavioral research.

[70]  George R. Franke,et al.  Structural Equation Modeling with EQS and EQS/Windows , 1995 .

[71]  Gerhard Arminger,et al.  Specification and Estimation of Mean Structures: Regression Models , 1995 .

[72]  T. Reilly,et al.  Equality in constraints and metric‐setting measurement models , 1995 .

[73]  Peter Ph. Mohler,et al.  Trends and Perspectives in Empirical Social Research , 1995 .

[74]  P M Bentler,et al.  A two-stage estimation of structural equation models with continuous and polytomous variables. , 1995, The British journal of mathematical and statistical psychology.

[75]  Clifford C. Clogg,et al.  Handbook of statistical modeling for the social and behavioral sciences , 1995 .

[76]  G. Arminger,et al.  Specification and Estimation of Mean- and Covariance-Structure Models , 1995 .

[77]  Sadanori Konishi,et al.  Application of the bootstrap methods in factor analysis , 1995 .

[78]  S. West,et al.  The robustness of test statistics to nonnormality and specification error in confirmatory factor analysis. , 1996 .

[79]  R. Hoyle Structural equation modeling: concepts, issues, and applications , 1997 .

[80]  Ke-Hai Yuan,et al.  Mean and Covariance Structure Analysis: Theoretical and Practical Improvements , 1997 .